Linear Algebra-Linear Subspaces?
For each of the following subsets of R2, indicate whether or not it is a
linear subspace of R2? Explain your reasoning in each case.
(i) {(x1, x2) : x1 + x2^2 = 0}
(ii) {(x1, x2) : x1 + x2 = 0}
(iii) {(x1, x2) : x1 + x2 = 1}
Id appreciate if someone could help! Im not sure at all whats meant by Linear Subspaces!
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(i) This is not a vector subspace, because it is not closed under (vector) addition.
For instance, (-1, 1) and (-1, -1) are in this set, but their sum (-2, 0) is not in it,
because (-2) + 0^2 ≠ 0.
(ii) This is a vector subspace; check the two closure axioms.
Let (a, b) and (c, d) be in this set (call it S).
So, a + b = 0 and c + d = 0.
Closure under vector addition:
(a, b) + (c, d) = (a+c, b+d) is in S, because
(a+c) + (b+d) = (a + b) + (c + d) = 0 + 0 = 0, as required.
Closure under scalar multiplication:
Letting k be a scalar in R, we see that k(c, d) = (kc, kd) is in S, because
kc + kd = k(c + d) = k * 0 = 0, as required.
(iii) This is not a vector subspace, because (0, 0) is not in the set. [0 + 0 ≠ 1.]
I hope this helps!